Bending Moment: Definition, Formula, and Bending Moment Diagrams (BMD)

What is a bending moment (in plain statics)

Take any beam under load. Pick a location x and imagine making a clean cut through the beam.

On that cut, the internal actions that keep equilibrium show up as resultants. In the simplest beam model you’ll see:

• Shear force \( V(x) \)
• Bending moment \( M(x) \)

The bending moment is the internal “turning” couple needed so that the left part and right part of the beam can balance the loads and stay in equilibrium.

Units

A bending moment is force × distance, so typical units are:

• N·m (often kN·m) in SI
• lb·ft in US customary

Sign convention (positive vs negative bending moment)

Most structural texts use:

• Positive moment = sagging (beam “smiles”): top fibers in compression, bottom fibers in tension
• Negative moment = hogging (beam “frowns”): top fibers in tension, bottom fibers in compression

This matters because software and textbooks can flip signs depending on axis definitions and conventions. The safe rule is boring but effective: define your sign convention once and keep it consistent from reactions → SFD → BMD → design checks.

How to calculate bending moment

For typical statically determinate beams, this is just equilibrium.

1. Compute support reactions.
2. Make a cut at position x.
3. Take one side of the cut and write moment equilibrium about the cut.
4. Solve for \( M(x) \).

A quick reminder of what you’re doing mathematically:

\( M(x) = \sum \big(F \cdot \text{lever arm}\big) + \sum \big(\text{applied moments}\big) \)

Example 1 — simply supported beam with uniform load (UDL)

Span L = 6 m, uniform load w = 4 kN/m.

Reactions:

\( R_A = R_B = \frac{wL}{2} = \frac{4\cdot 6}{2} = 12\ \text{kN} \)

Shear (left to right):

\( V(x) = 12 – 4x \)

Moment (integrate shear, or do equilibrium on the cut):

\( M(x) = 12x – 2x^2\ \text{(kN·m)} \)

Where is the maximum moment? Where the shear crosses zero:

\( V(x)=0 \Rightarrow 12 – 4x = 0 \Rightarrow x = 3\ \text{m} \)

Maximum moment:

\( M_{max}=M(3)=12\cdot 3 – 2\cdot 3^2 = 36 – 18 = 18\ \text{kN·m} \)

That’s also the well-known closed form:

Example 2 — simply supported beam with a midspan point load

Span L, point load P at midspan.

Maximum moment occurs at midspan:

\( M_{max} = \frac{PL}{4} \)

Bending moment diagram (BMD): how to draw it, how to read it

A BMD is a plot of \( M(x) \) along the beam. The point of the diagram is not the art. It’s locating:

• maximum moment (often where bending stress is highest)
• sign changes (points of contraflexure, where curvature flips)
• regions that control code checks (bending resistance, stability modifiers, combined checks)

What shapes to expect (fast intuition)

You can predict the BMD shape without doing a full derivation:

• No distributed load \( w=0 \) → shear is constant → moment is linear
• Uniform distributed load \( w=\text{const} \) → shear is linear → moment is parabolic
• Point load → shear jumps → moment slope changes abruptly

The “sanity-check” relationships

For prismatic beam theory, the following are your best debugging tools:

\( \frac{dM}{dx} = V, \qquad \frac{dV}{dx} = -w \)

If your shear diagram is flat, your moment diagram should be a straight line. If your shear diagram is a straight line, your moment diagram should be a parabola.

Common bending moment formulas (max moment cheat sheet)

You’ll see these in quick checks, hand calcs, and as a way to validate solver output:

• Simply supported + UDL (w): \( M_{max} = \frac{wL^2}{8} \)
• Simply supported + midspan point load (P): \( M_{max} = \frac{PL}{4} \)
• Cantilever + end point load (P): \( M_{max} = PL \) at the fixed end
• Cantilever + UDL (w): \( M_{max} = \frac{wL^2}{2} \) at the fixed end

If you need a full library (two-point loads, partial UDLs, continuous beams), put it on a dedicated “beam formulas” page and link it from here. This page should stay focused on understanding + BMD.

Bending moment vs shear force vs torque

These get mixed up constantly:

• Shear V is the internal force transverse to the member axis (in the beam idealization).
• Bending moment M is the internal couple primarily associated with normal stress varying across the depth of the section.
• Torque T is a twisting moment about the member’s longitudinal axis (different stress state and different checks).

Moment and torque share units, but the physics and failure modes are not interchangeable.

In 3D members, bending is not one number. You usually have axial force N and two bending moments about the local axes (in-plane and out-of-plane).

Internal actions in a member: axial force (N), in-plane bending moment, and out-of-plane bending moment.

From moment → stress and deflection (what changes, what doesn’t)

Two practical truths that save time:

1. For statically determinate beams, \( M(x) \) depends on loads and supports, not on material stiffness.
2. Stiffness controls deformation and influences stress through section properties.

Bending stress from moment

For linear elastic bending of a prismatic beam section:

\( \sigma = \frac{My}{I} \)

• (I) is the second moment of area (moment of inertia)
• (y) is the distance from the neutral axis

Maximum stress occurs at the extreme fibers (largest \( |y| \).

Curvature/deflection connection

In classic beam theory (how (M), (E), and (I) connect is covered in structural properties in FEA):

\( \kappa \propto \frac{M}{EI} \)

So: a correct BMD can still produce wrong deflections if (E), (I), boundary conditions, element formulation, or units are wrong.

Bending moments in FEA (and what SDC Verifier checks in practice)

In real projects you usually don’t draw BMDs by hand. You extract moments from analysis output (see: FEM in structural analysis) and then run design checks.

Where engineers get burned is not (and it’s why built-in CAD simulation isn’t enough once you’re doing real verification work) “the formula.” It’s the plumbing.

Common FEA moment mistakes

• Local axes flipped (often a beam cross-section orientation problem) → My/Mz signs swap, end moments don’t match expectations.
• Unit mismatch → \( \text{N·mm} \) interpreted as \( \text{kN·m} \) (or vice versa). Everything looks “reasonable” until it fails spectacularly.
• Mixing result types → beam end moments vs shell stress resultants vs solid stresses.
• Wrong coordinate system → reporting global when the check assumes local (or the reverse).

How SDC Verifier fits

SDC Verifier is a verification layer: you bring in analysis results, then standardize checks and reporting (see: engineering verification reports).

Two workflow points that matter for bending-moment-driven checks:

1. Member orientation consistency
   • If member local axes are inconsistent, moment extraction and moment-gradient factors become garbage.
   • Use Beam Member Finder to catch orientation issues early.
2. Moment-gradient / end-moment ratio factors (code checks)
   • Many standards modify capacity based on the moment diagram shape (moment ratio, moment gradient).
   • The Moment Ratio Tool computes these factors from the moment distribution so you don’t hand-calc end-moment ratios, sign cases, or diagram permutations.
   • Typical examples: equivalent uniform moment factors (Eurocode-style) and lateral-torsional buckling modifiers (AISC-style).

Moment distribution cases and the corresponding k₍c₎ factor (function of end-moment ratio ψ), used in Eurocode-style moment-gradient checks.

If your code check depends on moment gradient, always verify three things before blaming the standard:

• the governing load case is actually the one you think it is
• the member local axes are consistent across the set
• the extracted moment diagram has the expected shape (linear/parabolic segments where it should)

Equivalent uniform moment factor βM for different moment diagrams (end moments ψ, lateral-load moments MQ, and combined cases).

FAQ

What is a bending moment?

A bending moment is the internal couple at a cross-section of a member required for equilibrium under external loads. It causes bending and is plotted as a bending moment diagram along the span.

What is the formula for bending moment?

At a section, it’s the algebraic sum of external moments about that section for one side of the cut: \( M=\sum(F \cdot \text{lever arm}) \) plus any applied moments.

What units are used for bending moment?

Common units are \( \text{N·m} \) (often \( \text{kN·m} \)) in SI and \( \text{lb·ft} \) in US customary units.

What is BMD and SFD?

SFD is the shear force diagram \( V(x) \). BMD is the bending moment diagram \( M(x) \). They are related by \( \frac{dM}{dx}=V \).

Where is the maximum bending moment?

For many common cases it occurs where the shear crosses zero (or at a boundary like a fixed end). For a simply supported beam with UDL, it’s at midspan.

Why is sign convention confusing?

Because different texts and software define positive curvature/moment differently depending on axis orientation. Pick one convention and keep it consistent across diagrams, extraction, and checks.